HOLOMORPHIC MORSE INEQUALITIES AND SYMPLECTIC REDUCTION

We introduce Morse-type inequalities for a holomorphic circle action o n a holomorphic vector bundle over a compact Kahler manifold. Our ineq ualities produce bounds on the multiplicities of weights occurring in the twisted Dolbeault cohomology in terms of the data of the fixed poi nts and of the symplectic reduction. This result generalizes both the Wu-Zhang extension of Witten's holomorphic Morse inequalities and the Tian-Zhang Morse-type inequalities for symplectic reduction. As an app lication we get a new proof of the Tian-Zhang relative index theorem f or symplectic quotients. (C) 1998 Elsevier Science Ltd. All rights res erved.